This question may seem strange, but here is the idea.
We know that $\mathbb R$ is uncountable. We also picture $\mathbb R$ as a line with its points "continuously" placed, in contrast with a set like $\mathbb N$ which is made of "isolated" points.
Then we could believe that uncountability and continuity are the same, but they are not : the set $\mathbb R\setminus\mathbb Q$ of irrationals is also uncountable, but is contains no "continuous" interval, so is also made of "isolated" points. Then, what justifies us in saying that $\mathbb R$ is a continuous set ?
I guess topology has a say in this, so I anticipate with a second question : could we equip $\mathbb N$ or $\mathbb R\setminus\mathbb Q$ with a topology that makes it "continuous" like $\mathbb R$ intuitively is ?
The "continuity" of $\mathbb{R}$ is for me a consequence of the fact that it is both order dense (between each $x < y$ we can find a $z$ with $x < z < y$) and order complete: if $A \subseteq \mathbb{R}$ has an upper bound, then it has a least upper bound $\sup A$. Topologically these ensure that the reals in the order topology is locally compact (no "small holes") and connected, things that the irrationals, rationals and integers do not have.